When atoms are exposed to an intense femtosecond laser field, a wealth of exciting phenomena can be observed, including high-harmonic generation and above-threshold ionization. It is an important task to find out the temporal characteristics of the propagating intense femtosecond laser pulse for proper understanding of relevant physical phenomena. Among the ionization models for strong field phenomena, the ionization rates of Ammosov-Delone-Krainov and Perelomov-Popov-Terente`v models are compared with that obtained with numerically solved schrOdinger equation. An one-dimensional propagation model, including plasma and Kerr effects, is developed in order to calculate the spectral change of the intense femtosecond laser pulse transmitted through an ionizing gas. The spectrum is blueshifted and broadened due to the self-phase modulation of the laser pulse propagating in an ionizing gas medium. Spectrum of the laser pulse is affected by traversing distance, gas pressure, and laser intensity. Especially, it depends on the number of gas atoms in the beam path, i.e., traversing distance times gas pressure. The spectrum is broadened due to self-phase modulation resulting from the Kerr effect. Using the short-time Fourier transform, the spectral change of intense pulses including the focusing effect is analyzed. The spectral modulation observed in the spectral structure of the propagated femtosecond laser pulse matches well with the interpretation based on the spectral interferometry and in the case of a focused intense beam the femtosecond laser pulse develops a large positive chirp in the leading edge.
